A single purchase at the weighted average price is the same as the DCA'd purchase. In both cases you buy X shares for Y$.
The time-weighted average price was $37/share.
DCA is a way to force yourself to make purchases at a variety of prices and times. (How likely is it with a single-purchase strategy that you'd own 111 shares at $2.70/sh of a stock that charted $10->$100->$1? How would you feel when you saw your $300 investment go to $111 in value? DCA takes a lot of that psychological pressure off and is amenable to decide-once and automate.)
Yes, that exactly. It's a psychological trick (if you have all the money upfront) and not a financially sound one.